Biophotonics_Concepts_to_Applications

reflection when tanh 1 =n 2 =n 1 :(b) Show thath 1 þh 2 ¼ 90 at the Brewster

angle.

2 :8 Consider the perpendicular and parallelreflection coefficientsrxand ry, given

by Eqs. (2.25) and (2.26), respectively. By using Snell’s law from Eq. (2.22)

and the identity sinðabÞ¼ðsinaÞðcosbÞðcosaÞðsinbÞ;eliminate the

dependence on n 1 and n 2 to write rxand ryas functions ofθ 1 andθ 2 only.

That is, show that this yields

rx¼r?¼

sinðÞh 1 h 2

sinðÞh 1 þh 2

ry¼rjj¼

tanðÞh 1 h 2

tanðÞh 1 þh 2

2 :9 Consider the case when light traveling in air (nair= 1.00) is incident per-
pendicularly on a smooth tissue sample that has a refractive index ntissue=
1.50. (a) Show that the reflection and transmission coefficients are 0.20 and
0.80, respectively. (b) Show that the reflectance and transmittance values are
R = 0.04 and T = 0.96.
2 :10 Show that the reflection coefficients rxand ry, given by Eqs. (2.25) and (2.26)
can be written in terms of only the incident angleθ 1 and the refractive index
ratio n 21 =n 2 /n 1 as

r?¼rx¼

cosh 1  n^221 sin^2 h 1

(^1) = 2
cosh 1 þ n^221 sin^2 h 1
(^1) = 2
rjj¼ry¼
n^221 cosh 1  n^221 sin^2 h 1
(^1) = 2
n^221 cosh 1 þ n^221 sin^2 h 1
(^1) = 2
2 :11 Consider a plane wave that lies in the plane of incidence of an air-glass
interface. (a) Show that the values of the reflection coefficients are rx=
−0.303 and ry= 0.092 if this lightwave is incident at 45° on the interface.
(b) Show that the values of the transmission coefficients are tx= 0.697 and
ty= 0.728. Let nair= 1.00 and nglass= 1.50.
2 :12 Consider the case of two monochromatic interfering lightwaves. Suppose the
intensities are such that I 2 =I 1. (a) If the phase differenceφ=2π, show that
the intensity I of the composite lightwave is 4I 1. (b) Show that the intensity
of the composite lightwave is zero whenφ=π.
2 :13 The frequency stability given byΔν/νcan be used to indicate the spectral
purity of a light source. Consider a Hg^198 low-pressure lamp that emits at a
wavelength 546.078 nm and has a spectral bandwidthΔν= 1000 MHz.
(a) Show that the coherence time is 1 ns. (b) Show that the coherence length
is 29.9 cm. (c) Show that the frequency stability is 1.82× 10 −^6. Thus the
frequency stability is about two parts per million.
50 2 Basic Principles of Light